Complexity and Control: Towards a Rigorous Behavioral Theory by Vladimir G Ivancevic, Darryn J Reid

By Vladimir G Ivancevic, Darryn J Reid

The publication Complexity and keep watch over: in the direction of a Rigorous Behavioral concept of advanced Dynamical structures is a graduate-level monographic textbook, meant to be a singular and rigorous contribution to trendy Complexity concept.

This e-book includes eleven chapters and is designed as a one-semester path for engineers, utilized and natural mathematicians, theoretical and experimental physicists, laptop and fiscal scientists, theoretical chemists and biologists, in addition to all mathematically knowledgeable scientists and scholars, either in and academia, attracted to predicting and controlling complicated dynamical structures of arbitrary nature.

Readership: expert and researchers within the box of nonlinear technology, chaos and dynamical and intricate structures.

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Extra info for Complexity and Control: Towards a Rigorous Behavioral Theory of Complex Dynamical Systems

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K=−∞ z=a However, if a = ∞, then the Laurent expansion ∞ f (z) = ak k=−∞ 1 zk gives Res f (z) = −a−1 . z=∞ Here we give several examples of residues, calculated using computer algebra systems Reduce and M athematica: 7 A winding number is an integer which counts how many times the curve winds around the point a. October 10, 2014 14 11:9 Complexity and Control 9in x 6in b1966-ch02 2 Local Geometrical Machinery for Complexity and Control z 1 = , 2−2 z 2 z= 2 sin(z) = 1, Res z=0 z2 Res √ Res √ z= 2 Res z=2 pole order = 1, √ sin(z) sin[ 2] √ , = pole order = 1, 2 z −2 2 2 1 = −m, (z − (z − 2)2 Res [tan(z)] = −1, 1)m z=π/2 Res z=π/2 pole order = 1, tan(z) + sec(z) = −2, sec(z − π/2) pole order = 2, pole order = 1, pole order = 1.

An example of a bounded set is a set S of real numbers that is bounded (from above) if there is a real number k such that k ≥ s (∀s ∈ S). October 10, 2014 24 11:9 Complexity and Control 9in x 6in b1966-ch02 2 Local Geometrical Machinery for Complexity and Control The (main) K¨ ahler form ω is defined on M as a closed (dω = 0) and positive (ω > 0) exterior (1,1)-form,16 given (in local holomorphic coordinates z1 , · · · , zn 17 of an open chart U ⊂ M ) by: ω = igij dz i ∧ dz j , such that the corresponding K¨ ahler metric g is a positive and symmetric (1,1)-form: g = igij dz i ⊗ dz j .

8) is beginning to realize our big picture of the K¨ ahler behavioral geometrodynamics:     Symplectic Geo Riemannian Geo K¨ ahler .  + i   T ∗ M with T M with BGD Hamiltonian Dyn Lagrangian Dyn In the next section, we will give a proper definition of a K¨ ahler n-manifold. 3 From Kalman Systems to Riemann Manifolds In this section we will show how the (left) Riemannian component of the K¨ ahler dynamics naturally rises from our everyday work performed say with Matlab/Simulink. In the following section, we will do a similar modeling exercise with the (right) symplectic component of the general K¨ahler dynamics.

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